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December  2018, 38(12): 6327-6350. doi: 10.3934/dcds.2018270

Weak solvability of fractional Voigt model of viscoelasticity

Victor Zvyagin , and  Vladimir Orlov

Laboratory of Mathematical Fluid Dynamics, Voronezh State University, Universitetskaya pl., 1, Voronezh 394 018, Russia

* Corresponding author: Victor Zvyagin

To Professor Rafael de la Llave

Received  September 2017 Revised  April 2018 Published  September 2018

Fund Project: This research was supported by the Ministry of Education and Science of the Russian Federation (grant 14.Z50.31.0037) and by the Russian Foundation for Basic Research, project no. 16-01-00370.

In the present paper we establish the existence of weak solutions to one fractional Voigt type model of viscoelastic fluid. This model takes into account a memory along the motion trajectories. The investigation is based on the theory of regular Lagrangean flows, approximation of the problem under consideration by a sequence of regularized Navier-Stokes systems and the following passage to the limit.

Keywords: Viscoelastic fluid, a priori estimate, weak solution, approximative recurrences, regular Lagrangian flow.
Mathematics Subject Classification: Primary: 76A05; Secondary: 35Q35.
Citation: Victor Zvyagin, Vladimir Orlov. Weak solvability of fractional Voigt model of viscoelasticity. Discrete & Continuous Dynamical Systems, 2018, 38 (12) : 6327-6350. doi: 10.3934/dcds.2018270
References:
[1]

L. Ambrosio, Transport equation and Cauchy problem for $BV$ vector fields, Invent. Math., 158 (2004), 227-260.  doi: 10.1007/s00222-004-0367-2.  Google Scholar

[2]

M. Caputo and F. Mainardi, Linear models of dissipation in inelastic solidss, La Rivista del Nuovo Cimento, 1 (1971), 161-198.  doi: 10.1007/BF02820620.  Google Scholar

[3]

M. Caputo and F. Mainardi, A new dissipation model based on memory mechanism, Pure Appl. Geophys., 91 (1971), 134-147.  doi: 10.1007/BF00879562.  Google Scholar

[4]

D. Cordoba, C. Fefferman and R. de la Llave, On squirt singularities in hydrodynamics, SIAM J. Math. Anal., 36 (2000), 204-213.  doi: 10.1137/S0036141003424095.  Google Scholar

[5]

G. Crippa and C. de Lellis, The regularity results for diPernaions flow, J. Reine Angew. Math., 616 (2008), 15-46.  doi: 10.1515/CRELLE.2008.016.  Google Scholar

[6]

G. Crippa, Ordinary differential equations with non-Lipschitz vector fields, Boll. Union Mat. Ital., 1 (2008), 333-348.   Google Scholar

[7]

R. J. DiPerna and P. L. Lions, Ordinary differential equations, transport theory and Sobolev spaces, Invent. Math., 98 (1989), 511-547.  doi: 10.1007/BF01393835.  Google Scholar

[8]

A. N. Gerasimov, A generalization of linear laws of deformation and its application to objectives of internal friction, Prikl. Mat. Mech., 12 (1948), 251-260.   Google Scholar

[9]

I. Gyarmati, Non-equilibrium Thermodynamics, Field Theory and Variational Principles, Springer-Verlag, Berlin-Heidelberg-New York, 1970. doi: 10.1007/978-3-642-51067-0.  Google Scholar

[10]

A. A. Kilbas, H. M. Srivastava and J. J. Trujillo, Theory and Applications of Fractional Differential Equations, North-Holland Mathematics Studies. Vol. 204, Elsevier, Amsterdam, 2006.  Google Scholar

[11]

R. de la Llave and R. Obaya, Regularity of the composition operator in spaces of H$\rmö$lder functions, Discrete Contin. Dyn. Syst. Ser. A, 5 (1999), 157-184.   Google Scholar

[12]

F. Mainardi and G. Spada, Creep, relaxation and viscosity for basic fractional models in rheology, Eur. Phys. J. Spec. Top., 193 (2011), 133-160.  doi: 10.1140/epjst/e2011-01387-1.  Google Scholar

[13]

E. N. Ogorodnikov, V. P. Radchenko and N. S. Yashagin, Rheological model the viscoelastic body with memory and differential equations of fractional oscillators, Vestn. Samar. Gos. Tekhn. Univ., Ser. Fiz.-Mat. Nauki, 22 (2011), 255-268.   Google Scholar

[14]

V. P. Orlov and P. E. Sobolevskii, On mathematical model of viscoelasticity with a memory, Differ. Integral Equ., 4 (1991), 103-115.   Google Scholar

[15]

A. P. Oskolkov, On some quasilinears systems occuring in studing of motion of viscous fluids, Zap. Nauchn. Sem. LOMI, 52 (1975), 128-157.   Google Scholar

[16]

S. G. Samko, A. A. Kilbas and O. I. Marichev, Integrals and Derivatives of Fractional Order and Some of Their Applications, Nauka I Tekhnika, Minsk, 1987.  Google Scholar

[17]

G. Scott-Blair, Survey of General and Applied Rheology, 2$^{nd}$ edition, Isaac Pitman and Sons, London, 1949. Google Scholar

[18]

P. E. Sobolewskii, On equations of parabolic type in a Banach space, Trans. Moscow Math. Soc., 10 (1961), 297-350.   Google Scholar

[19]

R. Temam, Navier-Stokes Equation, North Holland Publishing Company, Amsterdam-New York-Oxford, 1979.  Google Scholar

[20]

V. G. Zvyagin and V. T. Dmitrienko, On weak solutions of regularized model of a viscoelastic fluid, (Russian)Dokl. Akad. Nauk, 380 (2001), 308-311. doi: 10.1023/A:1023860129831.  Google Scholar

[21]

V. G. Zvyagin and V. T. Dmitrienko, On weak solutions of an initial-boundary value problem for the equation of motion of a viscoelastic fluid, Dokl. Math., 64 (2001), 190-193.   Google Scholar

[22]

V. G. Zvyagin and M. V. Turbin, The study of initial-boundary value problems for mathematical models of the motion of Kelvin-Voigt fluids, J. Math. Sci., 168 (2010), 157-308.  doi: 10.1007/s10958-010-9981-2.  Google Scholar

[23]

V. G. Zvyagin and D. A. Vorotnikov, Approximation Methods for Evolutionary Problems of Nonlinear Hydrodynamics, De Gruyter Series in Nonlinear Analysis and Applications, Walter de Gruyter, Berlin-New York, 2008. doi: 10.1515/9783110208283.  Google Scholar

show all references

References:
[1]

L. Ambrosio, Transport equation and Cauchy problem for $BV$ vector fields, Invent. Math., 158 (2004), 227-260.  doi: 10.1007/s00222-004-0367-2.  Google Scholar

[2]

M. Caputo and F. Mainardi, Linear models of dissipation in inelastic solidss, La Rivista del Nuovo Cimento, 1 (1971), 161-198.  doi: 10.1007/BF02820620.  Google Scholar

[3]

M. Caputo and F. Mainardi, A new dissipation model based on memory mechanism, Pure Appl. Geophys., 91 (1971), 134-147.  doi: 10.1007/BF00879562.  Google Scholar

[4]

D. Cordoba, C. Fefferman and R. de la Llave, On squirt singularities in hydrodynamics, SIAM J. Math. Anal., 36 (2000), 204-213.  doi: 10.1137/S0036141003424095.  Google Scholar

[5]

G. Crippa and C. de Lellis, The regularity results for diPernaions flow, J. Reine Angew. Math., 616 (2008), 15-46.  doi: 10.1515/CRELLE.2008.016.  Google Scholar

[6]

G. Crippa, Ordinary differential equations with non-Lipschitz vector fields, Boll. Union Mat. Ital., 1 (2008), 333-348.   Google Scholar

[7]

R. J. DiPerna and P. L. Lions, Ordinary differential equations, transport theory and Sobolev spaces, Invent. Math., 98 (1989), 511-547.  doi: 10.1007/BF01393835.  Google Scholar

[8]

A. N. Gerasimov, A generalization of linear laws of deformation and its application to objectives of internal friction, Prikl. Mat. Mech., 12 (1948), 251-260.   Google Scholar

[9]

I. Gyarmati, Non-equilibrium Thermodynamics, Field Theory and Variational Principles, Springer-Verlag, Berlin-Heidelberg-New York, 1970. doi: 10.1007/978-3-642-51067-0.  Google Scholar

[10]

A. A. Kilbas, H. M. Srivastava and J. J. Trujillo, Theory and Applications of Fractional Differential Equations, North-Holland Mathematics Studies. Vol. 204, Elsevier, Amsterdam, 2006.  Google Scholar

[11]

R. de la Llave and R. Obaya, Regularity of the composition operator in spaces of H$\rmö$lder functions, Discrete Contin. Dyn. Syst. Ser. A, 5 (1999), 157-184.   Google Scholar

[12]

F. Mainardi and G. Spada, Creep, relaxation and viscosity for basic fractional models in rheology, Eur. Phys. J. Spec. Top., 193 (2011), 133-160.  doi: 10.1140/epjst/e2011-01387-1.  Google Scholar

[13]

E. N. Ogorodnikov, V. P. Radchenko and N. S. Yashagin, Rheological model the viscoelastic body with memory and differential equations of fractional oscillators, Vestn. Samar. Gos. Tekhn. Univ., Ser. Fiz.-Mat. Nauki, 22 (2011), 255-268.   Google Scholar

[14]

V. P. Orlov and P. E. Sobolevskii, On mathematical model of viscoelasticity with a memory, Differ. Integral Equ., 4 (1991), 103-115.   Google Scholar

[15]

A. P. Oskolkov, On some quasilinears systems occuring in studing of motion of viscous fluids, Zap. Nauchn. Sem. LOMI, 52 (1975), 128-157.   Google Scholar

[16]

S. G. Samko, A. A. Kilbas and O. I. Marichev, Integrals and Derivatives of Fractional Order and Some of Their Applications, Nauka I Tekhnika, Minsk, 1987.  Google Scholar

[17]

G. Scott-Blair, Survey of General and Applied Rheology, 2$^{nd}$ edition, Isaac Pitman and Sons, London, 1949. Google Scholar

[18]

P. E. Sobolewskii, On equations of parabolic type in a Banach space, Trans. Moscow Math. Soc., 10 (1961), 297-350.   Google Scholar

[19]

R. Temam, Navier-Stokes Equation, North Holland Publishing Company, Amsterdam-New York-Oxford, 1979.  Google Scholar

[20]

V. G. Zvyagin and V. T. Dmitrienko, On weak solutions of regularized model of a viscoelastic fluid, (Russian)Dokl. Akad. Nauk, 380 (2001), 308-311. doi: 10.1023/A:1023860129831.  Google Scholar

[21]

V. G. Zvyagin and V. T. Dmitrienko, On weak solutions of an initial-boundary value problem for the equation of motion of a viscoelastic fluid, Dokl. Math., 64 (2001), 190-193.   Google Scholar

[22]

V. G. Zvyagin and M. V. Turbin, The study of initial-boundary value problems for mathematical models of the motion of Kelvin-Voigt fluids, J. Math. Sci., 168 (2010), 157-308.  doi: 10.1007/s10958-010-9981-2.  Google Scholar

[23]

V. G. Zvyagin and D. A. Vorotnikov, Approximation Methods for Evolutionary Problems of Nonlinear Hydrodynamics, De Gruyter Series in Nonlinear Analysis and Applications, Walter de Gruyter, Berlin-New York, 2008. doi: 10.1515/9783110208283.  Google Scholar

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